I don't agree that those are the intervals that satisfy the inequality. Look at the graph again. You should see that one of the intervals for which [itex]\sin(x)<\sqrt{3}/2[/itex] is:

[tex]\left(\frac{2\pi}{3},\frac{7\pi}{3}\right)[/tex]

That interval isn't captured by any value of [itex]k[/itex] in your solution.

It will be useless to make union of all of them, right?

What makes you say that? The problem statement was not restricted to some subset of the real line. If you want to correctly answer the question then you must express it as the union of all of the intervals on which the inequality is satisfied.